Subgroups — #25067169

Order 11008.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145168	C₂	1	1	yes	maximal, characteristic
2	#19143572	C₂	1	2	yes	maximal, complemented
3	#19133508	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133504	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133500	C₂	1	2	yes	maximal, complemented
6	#19133420	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133405	C₂	1	2	yes	maximal, complemented
8	#19133355	C₂	1	1	yes	maximal, characteristic, complemented
9	#19133054	C₂	1	2	yes	maximal, complemented
10	#322×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583436	C₂²	1	2	yes	—
13	#14583300	C₄	1	1	yes	characteristic
14	#14583284	C₄	1	1	yes	characteristic
15	#14583255	C₄	1	2	yes	—
16	#14583137	C₄	1	1	yes	characteristic
17	#14583136	C₄	1	1	yes	characteristic
18	#14583131	C₄	1	2	yes	—
19	#14583030	C₂²	1	2	yes	—
20	C₄×#10621884	C₂²	1	2	yes	—
21	C₄×D₃₄₄	C₂²	1	1	yes	characteristic
22	#14582976	C₂²	1	2	yes	—
23	#14582969	—	2	2	no	—
24	C₈×D₁₇₂	—	2	2	no	—
25	C₈×D₁₇₂	C₂²	1	2	yes	—
26	#14582959	—	2	2	no	—
27	#14582955	C₂²	1	2	yes	—
28	#14582853	—	2	2	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
32	#14582651	C₂²	1	2	yes	—
33	#14582650	C₂²	1	1	yes	characteristic
34	Q₈⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
35	#14582628	C₂²	1	2	yes	—
36	#14582626	C₂²	1	2	yes	—
37	#14582621	C₂²	1	1	yes	characteristic
38	#14582621	C₂²	1	1	yes	characteristic
39	#14582620	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₄₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
42	C₄₃×#334	C₂²	1	1	yes	characteristic
43	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₄₃×#322	C₂²	1	1	yes	characteristic
45	C₂×#10621884	C₂×C₄	1	2	yes	—
46	C₂×D₃₄₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#10621872	—	2	2	no	—
48	C₂×C₈×D₄₃	—	2	1	no	—
49	C₂×C₈×D₄₃	—	2	2	no	—
50	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₄₃	D₄	1	2	yes	—
53	#13063320	—	2	2	no	—
54	#13063316	C₂×C₄	1	2	yes	—
55	#13063314	C₂×C₄	1	2	yes	—
56	#13063310	—	2	2	no	—
57	#13063301	C₂×C₄	1	2	yes	—
58	#13063300	C₂×C₄	1	2	yes	—
59	C₄×D₁₇₂	—	2	2	no	—
60	C₄×D₁₇₂	C₂³	1	2	yes	—
61	#13063282	—	2	2	no	—
62	C₄²×D₄₃	C₂³	1	1	yes	characteristic
63	#13063259	C₂×C₄	1	1	yes	characteristic
64	#13063256	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
66	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
67	#13063249	C₂×C₄	1	2	yes	—
68	#13063247	C₂×C₄	1	2	yes	—
69	#13063245	C₂×C₄	1	2	yes	—
70	(C₂×D₄₃)⋊C₈	—	2	2	no	—
71	(C₂×D₄₃)⋊C₈	—	2	2	no	—
72	(C₂×D₄₃)⋊C₈	—	2	2	no	—
73	#13063236	—	2	2	no	—
74	#13063235	—	2	2	no	—
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₄₃	—	2	1	no	—
79	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₃	D₄	1	2	yes	—
82	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
84	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₄₃	—	2	2	no	—
86	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
87	#103×C₄₃	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₄₃	—	2	2	no	—
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×D₁₇₂	—	2	2	no	—
94	C₂×D₁₇₂	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₃	—	2	2	no	—
96	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
99	Q₁₆×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
105	C₄⋊C₄×C₄₃	—	2	2	no	—
106	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₄₃	—	2	2	no	—
108	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
109	#10621884	—	2	4	no	—
110	(C₂×D₄₃)⋊C₄	—	2	2	no	—
111	(C₂×D₄₃)⋊C₄	—	2	2	no	—
112	(C₂×D₄₃)⋊C₄	—	2	2	no	—
113	(C₂×D₄₃)⋊C₄	—	2	2	no	—
114	(C₂×D₄₃)⋊C₄	—	2	2	no	—
115	C₄⋊Dic₄₃	—	2	2	no	—
116	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₃	#41	1	1	yes	characteristic
122	C₂×C₄₃⋊C₈	—	2	2	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
127	D₃₄₄	—	2	2	no	—
128	#10621872	—	4	4	no	—
129	C₈×D₄₃	—	2	2	no	—
130	C₈×D₄₃	—	2	2	no	—
131	C₈×D₄₃	—	4	2	no	—
132	C₈×D₄₃	—	4	4	no	—
133	C₈×D₄₃	—	4	2	no	—
134	C₂²×D₄₃	—	2	2	no	—
135	C₂²×D₄₃	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₄₃	—	2	2	no	—
137	Q₈×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	2	no	—
140	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₄₃	—	2	2	no	—
144	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
147	D₁₇₂	—	2	2	no	—
148	D₁₇₂	—	2	2	no	—
149	D₁₇₂	—	4	4	no	—
150	C₄×D₄₃	—	2	2	no	—
151	C₄×D₄₃	—	2	1	no	—
152	C₄×D₄₃	—	2	1	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	—	4	2	no	—
155	C₄×D₄₃	—	4	2	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₈×C₄₃	—	2	2	no	—
158	C₈×C₄₃	—	4	1	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	#131	1	2	yes	—
161	C₄₃⋊C₈	—	2	1	no	—
162	C₄₃⋊C₈	—	2	2	no	—
163	C₄₃⋊C₈	—	4	1	no	—
164	C₄₃⋊C₈	—	4	2	no	—
165	C₄₃⋊C₈	#131	1	2	yes	—
166	#17963	—	43	1	no	maximal, Sylow
167	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂×D₄₃	—	2	1	no	—
169	C₂×D₄₃	—	2	1	no	—
170	C₂×D₄₃	—	4	2	no	—
171	C₂×D₄₃	—	4	2	no	—
172	C₂×D₄₃	—	4	2	no	—
173	C₂×D₄₃	#444	1	2	yes	—
174	C₄×C₄₃	—	2	1	no	—
175	C₄×C₄₃	—	4	2	no	—
176	C₄×C₄₃	—	4	2	no	—
177	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
179	C₄×C₄₃	#575	1	2	yes	—
180	Dic₄₃	—	2	1	no	—
181	Dic₄₃	—	2	1	no	—
182	Dic₄₃	—	2	1	no	—
183	Dic₄₃	—	4	2	no	—
184	#3168	—	43	1	no	—
185	#2966	—	43	2	no	—
186	#1594	—	43	1	no	—
187	#1593	—	43	1	no	—
188	#1592	—	43	2	no	—
189	#1580	—	43	1	no	—
190	#1579	D₄₃	1	1	yes	characteristic, complemented
191	SD₁₆⋊C₈	—	43	2	no	—
192	#1568	—	43	1	no	—
193	#1494	—	43	2	no	—
194	C₂×#322	—	43	1	no	—
195	C₂×C₄₃	#2931	1	1	yes	characteristic
196	C₂×C₄₃	#2943	1	1	yes	characteristic
197	C₂×C₄₃	C₂×#444	1	1	yes	characteristic
198	D₄₃	—	2	2	no	—
199	D₄₃	—	8	2	no	—
200	#529	—	43	2	no	—
201	#499	—	43	1	no	—
202	#495	—	43	1	no	—
203	#488	—	43	2	no	—
204	#467	—	43	1	no	—
205	#466	—	43	1	no	—
206	#465	—	43	2	no	—
207	#446	—	43	2	no	—
208	C₈×Q₈	C₂×D₄₃	1	2	yes	—
209	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
210	C₄×SD₁₆	—	43	2	no	—
211	C₄×D₈	—	43	1	no	—
212	#436	—	43	2	no	—
213	#435	—	86	2	no	—
214	C₈×D₄	—	43	2	no	—
215	C₈×D₄	—	86	2	no	—
216	#433	—	86	2	no	—
217	#432	—	43	2	no	—
218	#405	—	86	2	no	—
219	C₂×C₄×C₈	—	43	1	no	—
220	C₂×C₄×C₈	—	43	1	no	—
221	C₂×C₄×C₈	—	43	1	no	—
222	#335	—	43	2	no	—
223	#334	—	43	1	no	—
224	#334	C₂×D₄₃	1	1	yes	characteristic
225	Q₈⋊C₈	—	43	2	no	—
226	Q₈⋊C₈	C₂×D₄₃	1	2	yes	—
227	D₄⋊C₈	—	43	2	no	—
228	D₄⋊C₈	—	43	2	no	—
229	#322	—	43	1	no	—
230	#322	—	43	1	no	—
231	#322	—	43	1	no	—
232	#322	C₂×D₄₃	1	1	yes	characteristic
233	C₄₃	#17963	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
235	C₂×SD₁₆	—	43	2	no	—
236	C₂×D₈	—	43	1	no	—
237	C₂×M₁₆	—	86	2	no	—
238	C₂²×C₈	—	43	1	no	—
239	C₂²×C₈	—	43	1	no	—
240	C₂²×C₈	—	43	2	no	—
241	C₂²×C₈	—	86	1	no	—
242	C₂²×C₈	—	86	2	no	—
243	#126	—	86	2	no	—
244	#125	—	43	2	no	—
245	#124	—	43	2	no	—
246	#123	—	86	2	no	—
247	C₂²⋊Q₈	—	43	2	no	—
248	#121	—	43	2	no	—
249	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
250	C₄×D₄	—	43	2	no	—
251	C₄×D₄	—	86	2	no	—
252	#117	—	86	2	no	—
253	C₂×C₄²	—	43	1	no	—
254	#107	—	43	1	no	—
255	#107	C₄×D₄₃	1	1	yes	characteristic
256	#106	—	43	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	43	2	no	—
259	C₄⋊C₈	—	86	2	no	—
260	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
261	#103	—	43	2	no	—
262	#103	C₄×D₄₃	1	2	yes	—
263	#102	—	43	2	no	—
264	#102	—	43	2	no	—
265	C₂²⋊C₈	—	86	2	no	—
266	C₂²⋊C₈	—	86	2	no	—
267	C₂²⋊C₈	—	86	2	no	—
268	#97	—	86	2	no	—
269	#97	—	86	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	43	1	no	—
272	C₄×C₈	—	43	1	no	—
273	C₄×C₈	—	43	2	no	—
274	C₄×C₈	—	43	1	no	—
275	C₄×C₈	—	43	1	no	—
276	C₄×C₈	—	43	1	no	—
277	C₄×C₈	—	86	1	no	—
278	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
281	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
282	C₂×D₄	—	43	2	no	—
283	C₂×D₄	—	86	2	no	—
284	C₂²×C₄	—	43	1	no	—
285	C₂²×C₄	—	43	1	no	—
286	C₂²×C₄	—	43	1	no	—
287	C₂²×C₄	—	86	2	no	—
288	Q₁₆	—	2	2	no	—
289	SD₁₆	—	86	4	no	—
290	D₈	—	86	2	no	—
291	M₁₆	—	172	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	43	1	no	—
295	C₂×C₈	—	43	1	no	—
296	C₂×C₈	—	43	2	no	—
297	C₂×C₈	—	86	2	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	—	86	2	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	172	2	no	—
302	C₂×C₈	—	172	4	no	—
303	C₂×C₈	—	172	2	no	—
304	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
306	C₂×C₈	D₄×D₄₃	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	43	2	no	—
309	C₄⋊C₄	—	86	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
313	C₂²⋊C₄	—	86	2	no	—
314	C₂²⋊C₄	—	86	2	no	—
315	C₂²⋊C₄	—	86	2	no	—
316	C₂²⋊C₄	—	86	2	no	—
317	C₂²⋊C₄	—	86	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	43	1	no	—
320	C₄²	—	43	1	no	—
321	C₄²	—	43	1	no	—
322	C₄²	C₂³×D₄₃	1	1	yes	characteristic
323	C₂³	—	43	1	no	—
324	C₂³	—	86	2	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	86	2	no	—
328	D₄	—	86	2	no	—
329	D₄	—	172	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	43	1	no	—
333	C₂×C₄	—	43	1	no	—
334	C₂×C₄	—	43	1	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	86	1	no	—
337	C₂×C₄	—	86	1	no	—
338	C₂×C₄	—	86	2	no	—
339	C₂×C₄	—	86	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	—	172	2	no	—
342	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	2	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	43	2	no	—
350	C₈	—	86	1	no	—
351	C₈	—	86	2	no	—
352	C₈	—	172	1	no	—
353	C₈	—	172	2	no	—
354	C₈	#13063331	1	2	yes	—
355	C₂²	—	43	2	no	—
356	C₂²	—	86	1	no	—
357	C₂²	—	86	1	no	—
358	C₂²	—	172	2	no	—
359	C₂²	—	172	2	no	—
360	C₂²	—	172	2	no	—
361	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	86	1	no	—
366	C₄	—	86	1	no	—
367	C₄	—	86	1	no	—
368	C₄	—	172	2	no	—
369	C₄	#131×D₄₃	1	1	yes	characteristic
370	C₄	C₂×#13063331	1	1	yes	characteristic
371	C₄	#14583698	1	2	yes	—
372	C₂	—	86	2	no	—
373	C₂	—	344	2	no	—
374	C₂	#19143302	1	1	yes	characteristic
375	C₂	#19143394	1	1	yes	characteristic
376	C₂	#444×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₄₃×#1579	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#14583136	C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	#1579	D₄₃
hypocenter	C₄₃	#17963
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #19143572	C₂	semidirect
3. #19133508	C₂	semidirect
4. #19133504	C₂	semidirect
5. #19133500	C₂	semidirect
6. #19133420	C₂	semidirect
7. #19133405	C₂	semidirect
8. #19133355	C₂	semidirect
9. #19133054	C₂	semidirect
10. #322×D₄₃	C₂	semidirect
11. C₄₃×#1579	C₂	semidirect
190. #1579	D₄₃	semidirect
233. C₄₃	#17963	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19145168	2
#19143572	2
#19133508	2
#19133504	2
#19133500	2
#19133420	2
#19133405	2
#19133355	2
#19133054	2
#322×D₄₃	2
C₄₃×#1579	2
#17963	43